resolve a cube into the sum of two cubes, a fourth power into two fourth powers, or in general any power higher than the second into two of the same kind, is impossible; of which fact I have found a remarkable proof. The margin is too small to contain it.”
This statement has come to be known as his “last theorem,” because for many years it was the only assertion of his that his successors had neither proved nor disproved. Nobody could reconstruct Fermat’s “remarkable proof,” and it seemed increasingly doubtful that he had really found one. But if he had possessed a proof, even though it couldn’t fit in a margin, it would surely be concise and elegant enough to earn a place in God’s Book? No one was writing blockbuster proofs in the seventeenth century. Yet for three and a half centuries, mathematician after mathematician failed to find Fermat’s missing proof. Then, in the late 1980s, Wiles began an extended attack on the problem. He worked alone in the attic of his house, telling only a few select colleagues who were sworn to secrecy.
Wiles’s strategy, like that of many mathematicians before him, was to assume that a solution existed and then play with the numbers algebraically in the hope that this would lead to a contradiction. His starting point was an idea emanating from the German mathematician Gerhard Frey, who realized that you could construct a type of cubic equation known as an elliptic curve from the three numbers occurring in the purported solution of Fermat’s “impossible” equation. This was a brilliant idea, because mathematicians had been playing around with elliptic curves for more than a century and had developed plenty of ways of manipulatingthem. What’s more, mathematicians then realized that the elliptic curve made from Fermat’s roots would have such strange properties that it would contradict another conjecture—the so-called Taniyama–Shimura–Weil conjecture—that governs the behavior of such curves.
No one had ever proved the Taniyama–Shimura–Weil conjecture, though most mathematicians thought it was probably right. If it were right, of course, the roots of Fermat’s equation would lead to a contradiction, showing that they could not exist. So Wiles took a deep breath and set about trying to prove the Taniyama–Shimura–Weil conjecture. For seven years, he brought every big gun of number theory to bear on it, until he eventually came up with a strategy that cracked it wide open. Although he worked alone, he didn’t invent the whole area by himself. He kept in close touch with all new developments on elliptic curves, and without a strong community of number theorists creating a steady stream of new techniques, he probably would not have succeeded. Even so, his contribution is massive, and it is propelling the subject into exciting new territory.
Wiles’s proof has now been published in full, and in print it comes to a bit over one hundred pages. Certainly too long to fit into a margin. Was it worth it?
Absolutely.
The machinery that Wiles developed to crack Fer-mat’s last theorem is opening up entire new areas ofnumber theory. Agreed, the story he had to tell was lengthy, and only experts in the area could hope to understand it in any detail, but it makes no more sense to complain about that than it would to complain that in order to read Tolstoy in the original, you have to be able to understand Russian.
12
Blockbusters
Dear Meg,
No, I was not joking when I said that the classification of the finite simple groups runs to ten thousand pages. It is currently being simplified and reorganized, though. With luck and a following wind, it might shrink to only two thousand. Most of the proof was carried out by hand, and the ideas behind it were entirely the product of the human mind. But some key parts required computer assistance.
This is a growing trend, and it has led to a new kind of narrative style for proofs, the computer-assisted proof, which has come
This statement has come to be known as his “last theorem,” because for many years it was the only assertion of his that his successors had neither proved nor disproved. Nobody could reconstruct Fermat’s “remarkable proof,” and it seemed increasingly doubtful that he had really found one. But if he had possessed a proof, even though it couldn’t fit in a margin, it would surely be concise and elegant enough to earn a place in God’s Book? No one was writing blockbuster proofs in the seventeenth century. Yet for three and a half centuries, mathematician after mathematician failed to find Fermat’s missing proof. Then, in the late 1980s, Wiles began an extended attack on the problem. He worked alone in the attic of his house, telling only a few select colleagues who were sworn to secrecy.
Wiles’s strategy, like that of many mathematicians before him, was to assume that a solution existed and then play with the numbers algebraically in the hope that this would lead to a contradiction. His starting point was an idea emanating from the German mathematician Gerhard Frey, who realized that you could construct a type of cubic equation known as an elliptic curve from the three numbers occurring in the purported solution of Fermat’s “impossible” equation. This was a brilliant idea, because mathematicians had been playing around with elliptic curves for more than a century and had developed plenty of ways of manipulatingthem. What’s more, mathematicians then realized that the elliptic curve made from Fermat’s roots would have such strange properties that it would contradict another conjecture—the so-called Taniyama–Shimura–Weil conjecture—that governs the behavior of such curves.
No one had ever proved the Taniyama–Shimura–Weil conjecture, though most mathematicians thought it was probably right. If it were right, of course, the roots of Fermat’s equation would lead to a contradiction, showing that they could not exist. So Wiles took a deep breath and set about trying to prove the Taniyama–Shimura–Weil conjecture. For seven years, he brought every big gun of number theory to bear on it, until he eventually came up with a strategy that cracked it wide open. Although he worked alone, he didn’t invent the whole area by himself. He kept in close touch with all new developments on elliptic curves, and without a strong community of number theorists creating a steady stream of new techniques, he probably would not have succeeded. Even so, his contribution is massive, and it is propelling the subject into exciting new territory.
Wiles’s proof has now been published in full, and in print it comes to a bit over one hundred pages. Certainly too long to fit into a margin. Was it worth it?
Absolutely.
The machinery that Wiles developed to crack Fer-mat’s last theorem is opening up entire new areas ofnumber theory. Agreed, the story he had to tell was lengthy, and only experts in the area could hope to understand it in any detail, but it makes no more sense to complain about that than it would to complain that in order to read Tolstoy in the original, you have to be able to understand Russian.
12
Blockbusters
Dear Meg,
No, I was not joking when I said that the classification of the finite simple groups runs to ten thousand pages. It is currently being simplified and reorganized, though. With luck and a following wind, it might shrink to only two thousand. Most of the proof was carried out by hand, and the ideas behind it were entirely the product of the human mind. But some key parts required computer assistance.
This is a growing trend, and it has led to a new kind of narrative style for proofs, the computer-assisted proof, which has come
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